1. THE SIMPLEX ALGORITHM Step 1 The objective row is scanned and the column containing the most negative term is selected (pivotal column) - indicate with an arrow. Step 2 Divide the value term in each row by the entry in the pivotal column (  -values). The pivot (element) is that which yields the least (non-negative) result. Indicate the pivotal row with an arrow and circle the pivot.

2. Step 3 Divide the whole pivotal row by the pivot (this leaves the number 1 in place of the pivot) and change the basic variables. Step 4 The aim now is to get zeros everywhere else in the selected column. This is done by adding to each row, multiples of the new pivotal row. The new rows form the next tableau. (You should now have zeros in the pivotal column except for a one where the pivot was) Step 5 Steps 1 to 4 are repeated until the objective row contains no negative elements. At this point, the objective has been attained.

3. Step 6 To “decode” the information, read down the “basic variable” column, the number in the value column gives the value of that variable. The value of all other variables not present in the final tableau is zero. This is the optimal solution. This will all appear meaningless until we illustrate it with an example

4. Maximise the function Introduce slack variables as follows: 2x + y + 1r + 0s = 32 x + y + 0r + 1s = 18 f – 80x – 70y + 0r + 0s = 0 This can be set out in a table as follows:

5. b.vxyrsValue r s f 2x + y + 1r + 0s = 32 211032 x + y + 0r + 1s = 18 110118 f - 80x - 70y + 0r + 0s = 0 -80-70000 Objective function Step 1 The objective row is scanned and the column containing the most negative term is selected (pivotal column) - indicate with an arrow. Step 2 Divide the value term in each row by the entry in the pivotal column (  -values). The pivot (element) is that which yields the least (non-negative) result. Indicate the pivotal row with an arrow and circle the pivot. 32/2 = 16 θ values 18/1 = 18 Step 3 Divide the whole pivotal row by the pivot (this leaves the number 1 in place of the pivot) and change the basic variables. 10.5 16 x Step 4 The aim now is to get zeros everywhere else in the selected column. This is done by adding to each row, multiples of the new pivotal row. The new rows form the next tableau. (You should now have zeros in the pivotal column except for a one where the pivot was) R2 – R100.5- 0.512 R3 + 80R10-304001280 Don’t do this for the objective row

6. We now have the following tableau: b.vxyrsValue x s f 1 10 0 0 0.5-0.5 0.5 2 0 16 -30401280 Step 1 The objective row is scanned and the column containing the most negative term is selected (pivotal column) - indicate with an arrow. Step 5 Steps 1 to 4 are repeated until the objective row contains no negative elements. At this point, the objective has been attained. Step 2 Divide the value term in each row by the entry in the pivotal column (  -values). The pivot (element) is that which yields the least (non-negative) result. Indicate the pivotal row with an arrow and circle the pivot. θ values 16/0.5 = 32 2/0.5 = 4 Step 3 Divide the whole pivotal row by the pivot (this leaves the number 1 in place of the pivot) and change the basic variables. 01- 124 Step 4 The aim now is to get zeros everywhere else in the selected column. This is done by adding to each row, multiples of the new pivotal row. The new rows form the next tableau. (You should now have zeros in the pivotal column except for a one where the pivot was) R1 - ½R2 10 y 114 R3 + 30R2 0010601400

7. We now have the following tableau: b.vxyrsValue x10114 y0124 f0010601400 Step 6 To “decode” the information, read down the “basic variable” column, the number in the value column gives the value of that variable. The value of all other variables not present in the final tableau is zero. This is the optimal solution. x = 14 y = 4 f = 1400 Also r = s = 0 Step 5 Steps 1 to 4 are repeated until the objective row contains no negative elements. At this point, the objective has been attained. No negatives